A pump device is already known from DE197 35 549 B4, having an electrical delivery pump, having a first port for connection to an air filter, having a second port for connection to a fuel tank, having a connecting line which connects the first port to the second port via a first valve opening, and having a bypass line which branches off from the connecting line upstream of the first valve opening and which opens into the connecting line downstream of the first valve opening via a throttled bypass outlet. Here, at the first valve opening, there is formed a first valve seat which interacts with a valve body arranged movably in a valve body receptacle. The pump device comprises an electrical switching valve which has an electrical actuator, for example a magnet coil, for moving the valve body.